Nov 17, 2009 1:31pm
Re: caution (do not step on tracks)
The Astrophysical Journal, 552:326-339, 2001 May 1
© 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.
Asteroseismological Constraints on the Structure of the ZZ Ceti Stars L19-2 and GD 165
P. A. Bradley 1
X-2, MS B220, Los Alamos National Laboratory, Los Alamos, NM 87545
Received 2000 October 11; accepted 2001 January 8
This study compares the theoretical pulsation periods from an extensive grid of evolutionary DA white dwarf models with the observed periods of the ZZ Ceti white dwarfs L19-2 and GD 165, in order to constrain their internal structure. Our analysis of the rotational fine-structure splitting and comparison of our theoretical periods with observations for L19-2 and GD 165 enable us to identify the observed modes as low-order ℓ = 1 and 2 g-modes. Because the period structure of GD 165 is quite similar to that of L19-2, we believe that the interior structure of GD 165 is similar. The short period of the ℓ = 1 118.5 s mode of L19-2 (120.4 s mode of GD 165) implies a hydrogen layer mass of about 10-4 M*, independent of constraints from the other pulsation modes. Detailed model fitting shows that L19-2 has a hydrogen layer mass of 1.0 × 10-4 M*, a helium layer mass of 1.0 × 10-2 M*, a 20 : 80 C/O core that extends out to 0.60 M*, a stellar mass of 0.72 M⊙, and a rotation period of about 13 hr. The best-fitting models for GD 165 have a hydrogen layer mass of 1.5 to 2.0 × 10-4 M*, a helium layer mass of 1.5 to 2.0 × 10-2 M*, a 20 : 80 C/O core that extends out to 0.65 M*, a stellar mass of 0.65–0.68 M⊙, and a rotation period of about 58 hr. In both cases, the best-fitting models are consistent with the spectroscopic log g-value, and the seismological parallax is within 1 σ of the observed parallax value.
Subject headings: stars: evolution; stars: individual (L19-2, GD 165); stars: oscillations; white dwarfs
The goal of asteroseismology is to derive the internal structure of pulsating stars by comparing the observed pulsation properties with those predicted by theoretical models and pulsation theory. ZZ Ceti stars are pulsating hydrogen atmosphere (DA) white dwarfs, and they hold the key to understanding the internal structure of the predominant spectral class of white dwarf. Until recently, there has been relatively little progress on ZZ Ceti star seismology, because the pulsation spectra in the hotter variables are so simple that there is no good way to choose from the many possible models that fit the observational data. On the other hand, cooler ZZ Ceti stars have more complicated pulsation spectra, with numerous harmonics (nf1) and frequency sum (f1+f2) and frequency difference (f1-f2) peaks due to the strongly nonlinear pulse shapes; many of these stars also have pulsation spectra that vary dramatically with time, further complicating mode identification.
Recently, people started making progress on the seismology of ZZ Ceti stars. Bradley (1998a) summarizes much of the recent developments for G226-29, GD 165, G117-B15A, and R548, which are hotter ZZ Ceti stars. In addition, Bradley (1998b) presents constraints on the structure of G117-B15A and R548 by comparing the observed periods to theoretically predicted periods from an extensive grid of evolutionary DA white dwarf models. Table 1 summarizes the recent structural constraints for the ZZ Ceti stars, with entries for the stellar mass, hydrogen layer mass, and rotation rate (where applicable). Table 1 shows hydrogen layer masses ranging from about 10-4 M* to 2 × 10-10 M*, although four of the six stars listed have values close to 10-4 M*. Both L19-2 and GD 165 are included in Table 1, but the previous seismology results are obsolete because of new (cooler) effective temperature and log g determinations.
Table 1 Previous Seismological Data for ZZ Ceti Stars
These results imply that the hydrogen layer mass of some pulsating DA white dwarfs lies between ~10-4 M* and ~10-10 M*. This hydrogen layer mass range is especially interesting because Clemens (1994) studied the pulsation properties of the hotter ZZ Ceti pulsators (including the previously mentioned stars) as a class and found that the observed periods are fitted quite well when the hydrogen layer mass is near 10-4 M*.
The one exception is GD 154, where Pfeiffer et al. (1996) give a preferred hydrogen layer mass 2 × 10-10 M*. Pfeiffer et al. find that models with hydrogen layers closer to 10-4 M* also fit, but then one has to invoke a highly selective mechanism that allows only three out of several dozen theoretically predicted modes to be observable. In contrast, Fontaine & Wesemael (1987, 1991) highlight a "thin" hydrogen layer (MH ≲ 10-10 M *) evolution path, which must apply to the stars that become DA white dwarfs in the "DB gap," the temperature range where no helium atmosphere white dwarf exists. Fontaine & Wesemael (1997) discuss the latest spectroscopic constraints on the hydrogen layer mass in the context of "spectral evolution," and they conclude that the DA white dwarfs must have a range of hydrogen layer masses ranging from "thick" at 10-4 M* to "thin" at 10-10 M*.
Given the preliminary state of present seismological efforts and the competing extreme hypotheses of Fontaine & Wesemael (1987, 1991) and Clemens (1994), it is critical to determine the structure of as many ZZ Ceti stars as possible to offer solid observational constraints. Based on the observational results of O'Donoghue & Warner (1982, 1987) and Sullivan et al. (2001) for L19-2 (WD 1425-811) and Bergeron et al. (1993) for GD 165 (WD 1422+095), we now have an observational database adequate for constraining their internal structure.
This paper examines in more detail the constraints that one can place on the structure of L19-2 and GD 165. For each star, we first see if we can fit a model with the mass as determined spectroscopically by Bergeron et al. (1995b) to the observed periods. Then we expand our model-fitting search to include models with other masses to see how well constrained the stellar mass, hydrogen layer mass, and other internal parameters are. L19-2 and GD 165 have three periods (112, 120, and 192 s) that are quite similar, and we start by assuming they have similar internal structures. We will test this assumption in our attempts to fit all the observed periods for each star.
2. MODELING STRATEGY AND CONSTRAINTS
We adopt an exhaustive approach to the analysis of these stars by considering a large grid of models that are a superset of the Bradley (1996) DA models. This grid allows a determination of the model structural properties (and uncertainties) comprehensively enough to constrain the structure of L19-2 and GD 165. Much of our present strategy is the same as that used by Bradley (1998b) to constrain the structure of R548 and G117-B15A. The major difference lies in our constraints on the identity of the observed modes, which we cover below.
2.1. Observational Mode Identification for L19-2
For L19-2, we use the pulsation periods of O'Donoghue & Warner (1982) and supplement them with the results of Sullivan et al. (2001); the combined list is shown in Table 2. Bradley (1993) discusses the possible mode identifications based on the periods from O'Donoghue & Warner (1982), and we follow that work here. L19-2 has five multiplets, and their periods are 113, 118, 143, 192, and 350 s. Three of these multiplets have spacings near 12 μHz, suggesting that they have a common ℓ-value. The 113 s mode has two peaks separated by 39.6 μHz, which is probably twice Δm = 19.8 μHz, because this better matches the 18.5 μHz splitting found in the 143 s mode, and the ratio of 12/19 is 0.63, close to the 0.60 predicted by asymptotic theory for the ratio of splitting between ℓ = 1 (12 μHz) and ℓ = 2 (19μ Hz) modes. Therefore, the frequency-splitting information suggests that the 118, 192, and 350 s modes are ℓ = 1, whereas the other two modes are ℓ = 2. The 118 and 192 s modes are obvious triplets, and the data set of Sullivan et al. (2001) suggests the presence of other peaks with frequency splittings consistent with those of O'Donoghue & Warner (1982) for the ℓ = 2 multiplets at 113 and 143 s. The fine-structure constraints lead to the m identification shown in Table 2. We choose m = 0 for the 350.15 s mode mainly because the other two ℓ = 1 multiplets have the m = 0 mode with the largest amplitude. There are two additional constraints available. Bradley (1993) points out that ℓ = 1 and ℓ = 2 modes with the same k have a period ratio close to 1.70, and this is matched by the 1.69 ratio between the 113 and 192 s periods, implying that they are ℓ = 2 and 1 modes with the same k-value. The shortness of the 118 s ℓ = 1 period implies that it must be the k = 1 mode, or we would have a hydrogen layer so thick that hydrogen burning at the base of the hydrogen layer would produce more luminosity than is observed. If the 118 s mode is k = 1, then the 113 and 192 s modes are probably k = 2, and we must rely on comparisons to models to determine the k-value of the 143 and 350 s modes. We will show that the 143 s mode is ℓ = 2, k = 4 and the 350 s mode is ℓ = 1, k = 6.
Table 2 Observed periods for L19-2 and GD 165
The mode identification for GD 165's periods is not as obvious as that for L19-2's. We have two probable triplets at 120 and 192 s, along with lower-amplitude modes near 114 and 250 s. Assuming the 120 and 192 s modes really are triplets, then these two periods are tantalizingly similar to those of L19-2, leading us to claim that these are the ℓ = 2 and ℓ = 1, k = 2 modes, respectively. The probable presence of a mode at 114 s in GD 165 reinforces this notion, and it would be the ℓ = 1, k = 1 mode, just like the 113 s mode of L19-2. The identity of the 250 s mode is unknown, but the ratio of 250/143 (from L19-2) is 1.75. This value is close enough to the 1.70–1.73 expected for ℓ = 1/ℓ = 2 period ratio to keep in mind; in this case, the 250 s period would be an ℓ = 1, k = 3 or 4 mode. As we will show, model fitting suggests that the 250 s mode is the ℓ = 1, k = 4 mode.
2.2. Teff and log g Constraints
We use the effective temperatures and surface gravities of Bergeron et al. (1995b) and their error bars to limit the size of the grid in effective temperature and mass. For convenience, the effective temperature, log g, and parallaxes of L19-2 and GD 165 are presented in Table 3. In addition, Koester & Holberg (2001) computed Teff- and log g-values using the shape of the UV spectrum, along with the added constraint of the V magnitude or the parallax (they use the same parallaxes we present in Table 1). Their values for L19-2 (Teff = 12,130 ± 90 K and log g = 8.17 ± 0.10) are essentially identical to our Table 3 values. The situation is less clear for GD 165. When Koester & Holberg (2001) use the V magnitude as the additional constraint, they derive Teff = 12,310 ± 180 K and log g = 8.20 ± 0.15. When the parallax is the additional constraint, then they derive Teff = 11,800 ± 330 K and log g = 7.76 ± 0.35. Given the large error bars, both results are formally the same as the values in Table 3. In § 7, we will see that we have trouble matching the observed parallax angle, and we would consider the V-magnitude constraint to be more reliable. We will comment on how the Koester & Holberg values compare to our GD 165 seismology results in § 7. The similarity in periods of the two stars suggests that the two should have similar masses and temperatures; however, Bergeron et al.'s (1995b) spectroscopic analysis suggests that GD 165 has a mass 0.09 M⊙ lower and a temperature slightly cooler than L19-2. Bergeron et al.'s log g-values suggest that the mass of L19-2 should lie between 0.71 and 0.80 M⊙, whereas GD 165 lies between 0.60 and 0.68 M⊙. For GD 165, the model temperature range lies between 11,700 and 12,300 K; the temperature range is 11,800–12,400 K for L19-2.
Table 3 Observational Data for L19-2 and GD 165
Koester & Vauclair (1997) show that the temperature and surface gravity uncertainties may be underestimated by χ2 minimization. One can vary the best-fitting effective temperature by up to 1000 K—with suitable changes to the surface gravity and convective efficiency—and obtain a formally identical fit to optical spectra. Only by considering the trigonometric parallax and/or the V magnitude is it possible to limit the luminosity range and, by extension, effective temperature errors. We compare the trigonometric parallax values cited in van Altena, Lee, & Hoffleit (1995) (included in Table 1) with the seismological parallaxes derived from our best-fitting models.
2.3. Determining the Best Model
To determine which model best fits the observed periods, we simply average the absolute value of the difference between the observed and theoretical periods and denote this average by Δ in the period comparison tables. More sophisticated comparisons are possible, but we feel that the discrete nature of the model grid and the small number of observed periods make the reward not worth the extra effort.
3. THE FREE PARAMETER QUESTION
There are nine potentially "free" parameters that one can adjust for modeling white dwarfs. The first four are (1) the effective temperature, (2) the surface gravity (total stellar mass), (3) the spherical harmonic indices (ℓ) of the pulsation modes, and (4) the overtone numbers (k) of the modes. We fix the effective temperature and stellar mass by means of the spectroscopic values provided by Bergeron et al. (1995b). The work of Bradley (1993) and the discussion in the previous section suggest mode identifications for both ℓ and k for most of the modes of L19-2 and GD 165. The mode identifications fix the first four free parameters. The hydrogen layer mass (the fifth parameter) is strongly constrained by requiring that the 118 s (or 120 s) mode be an ℓ = 1 mode. As mentioned above, this period is short enough that it must be the k = 1 mode. The sixth free parameter is the thickness of the base of the H/He transition zone; we find that the periods of several modes are sensitive to the transition zone extent. Our best-fitting models for both stars require adjustments to the "nominal" value. In our work, we have a "thick" (or "normal") transition region (see Fig. 1), and two thinner transition regions, which we label "moderately thin" and "thin." Bradley (1996) discusses the effect the thin and thick transition regions have on the periods (see also § 6.2). One consequence is that models with "thin" transition zones have shorter k = 1 periods for a given structure. The seventh and eighth parameters are the helium layer mass and the C/O core composition profile. These will be sensitive to the period distribution of the remaining modes after we constrain the hydrogen layer mass with the ℓ = 1, k = 1 period.
Fig. 1 Schematic diagram of the H/He (left) and He/C (right) transition regions, showing the difference between the "thick" (normal) transition H/He transition zone base and the "moderately thin" and "thin" transition zone bases.
The ninth parameter is the convective prescription, but the convective prescription changes only the extent of the convection zone. It is so near the surface that it has very little effect on the pulsation periods. Nonadiabatic pulsation studies by Bradley & Winget (1994) and Fontaine et al. (1994) show that convection must be efficient to match the observed instability strip, so we use their ML2 or ML3 convective efficiency.
Our observational constraints leave us with the overtone number, hydrogen layer mass, helium layer mass, and C/O core profile as the free parameters. Because the overtone number is strongly constrained for most of the modes, we can match the observed period distribution for only a limited number of models that are functions of effective temperature, hydrogen layer mass, and total stellar mass.
4. OVERVIEW OF WHITE DWARF MODEL BUILDING
We use models generated with the white dwarf evolution code discussed by Bradley (1996) and Wood (1994) in this analysis; the pulsation periods of some models are in Tables 1–12 of Bradley (1996). These models use a "nominal" C/O profile (see Fig. 1 of Bradley 1996) that is a simplified version of the oxygen-rich core profiles computed from evolutionary models by Mazzitelli & D'Antona (1986a, 1986b) and Salaris et al. (1997). For this work, the core is 80% oxygen out to 0.55–0.80 M*, at which point there is a linear change to pure carbon by 0.90 M*. We study helium layer masses between 3 × 10-2 M* and 6 × 10-3 M*, for reasons that we explain in § 6.2. We use a slightly modified version of the adiabatic nonradial pulsation analysis code described by Kawaler, Hansen, & Winget (1985). Bradley (1996) discusses the application of this Runge-Kutta-Fehlberg code to DA white dwarf models, and Bradley (1993) presents a preliminary application of matching DA models with the then favored temperature of 13,300 K to L19-2.
We compute only adiabatic periods for seismology for two reasons. First, the differences between the adiabatic and nonadiabatic periods are small, typically on the order of 0.01%. Adiabatic period calculations are much faster, and the many modes we calculate make us favor the extra speed. Second, the location of the high-temperature boundary of the instability strip and the spectrum of unstable modes for a given effective temperature are dominated by the choice of convective efficiency (Bradley & Winget 1994). Because the convective efficiency is a free parameter, we are free to adjust it until the models have only the first few modes unstable between 11,500 and 12,000 K. Therefore, we choose the ML2 or ML3 prescriptions of Bradley & Winget (1994). However, the choice of convective efficiency has a negligible effect on the pulsation periods, so we are not concerned about it for period matching.
5. STRUCTURAL CONSTRAINTS FOR L19-2 AND GD 165 FROM THE 118 SECOND (OR 120 SECOND) PERIOD
We can constrain the hydrogen layer mass fairly well for L19-2 and GD 165 because the 118 s (or 120 s) period is almost certainly the ℓ = 1, k = 1 mode (see § 3). This mode identification also has the benefit of producing the thinnest possible hydrogen layer mass for a given stellar mass. First, we look at a series of DA models near 12,100 K with differing stellar mass and hydrogen layer mass, keeping the rest of the composition profile fixed. In this section, we use the 118 s period alone to constrain the hydrogen layer mass as a function of stellar mass values. Figure 2 shows the trend of pulsation periods for some low-overtone ℓ = 1 modes from ~12,100 K models of 0.65, 0.70, 0.75, and 0.80 M⊙ with the nominal C/O profile. The theoretical k = 1 periods increase steeply with decreasing hydrogen layer mass, and Figure 2 shows that the inferred hydrogen layer mass ranges from 2 × 10-4 M* to 10-5 M* for stellar masses between 0.70 and 0.80 M⊙. Even when the hydrogen layer mass is the unreasonably thick value of 3 × 10-4 M*, the 0.65 M⊙ model has an ℓ = 1, k = 1 period of 122 s. This period is already several seconds longer than the 118 s (120 s) mode, so it is unlikely that L19-2 has this low a stellar mass.
Fig. 2 Periods for the first two-overtone ℓ = 1 modes of models near 12,100 K at 0.65, 0.70, 0.75, and 0.80 M⊙. Horizontal lines represent the periods for L19-2 (dashed) and GD 165 (dot-dashed). The 192 s mode for both stars has a nearly identical period. The best-fitting hydrogen layer mass is about 10-4 M* according to the figure.
We also plot periods for the ℓ = 1, k = 2 mode in Figure 2, and for this set of model parameters, the closest match to the observed 192 s period comes when the stellar mass is 0.70 M⊙ and the hydrogen layer mass is between 1.5 × 10-4 M* and 1.0 × 10-4 M*. We will keep these "first look" model parameters in mind when we attempt to fit all of the observed periods with our models.
6. STRUCTURAL CONSTRAINTS FROM ALL PERIODS FOR L19-2
From Figure 2, we see that fitting the 118 s period constrains the hydrogen layer mass to be near 10-4 M* and suggests that the stellar mass is about 0.70 M⊙. We have four other rotationally split modes for L19-2, listed in Table 2, that we can also compare with our model predictions. We perform our analysis in several parts; first, we discuss the period fitting for 0.70 and 0.75 M⊙ models, then we discuss the quality of the matches and the resultant chemical composition profile. We then discuss the rotation periods derived from our calculated fine-structure splitting constants. After the detailed discussion, we briefly discuss our results for models below 0.70 M⊙ and then for models with masses of 0.72, 0.74, and 0.78 M⊙. While this model grid does not exhaust all of the possible models that might fit L19-2, the grid is dense enough to determine the trend of Teff, MH, MHe, and the chemical composition profile as a function of stellar mass. The number of models we compute for a given stellar mass varies. Our original model grid had models of 0.70, 0.75, and 0.80 M⊙. We computed large numbers of models for these masses. Later, we looked at the intermediate masses after we already had a good idea of the chemical composition profile that would produce the best period match. Therefore, we computed relatively few of these models.
In addition to the period matches, we use the m-splitting information as an additional check on the validity of our model fit and compute the implied rotational period. To compute the rotational period, we use the observed m splittings and the first-order rotational splitting formula
and solve for Ω = 1/Prot. Ideally, the different observed frequency splittings and Cℓ, k-values should produce the same rotation period for each mode if the star is uniformly rotating. One trend we note from all the models is that the rotation periods implied by the 118, 192, and 114 s periods can be quite similar to each other and are about 13 hr. The frequency split between the 348.7 and 350.2 s modes is 11.62 μHz. O'Donoghue & Warner (1982) show that the 348.7 s mode is not part of the alias pattern of the 350.2 s mode, although we cannot rule out the possibility of a frequency shift due to the 1 cycle per day alias at 11.57 μHz. Our models almost always yield a Cℓ, k-value of 0.490–0.495, and this Cℓ, k-value implies a rotation period of about 12.2 hr. Alternatively, if we take the C1, 6-value of 0.490 as correct, the splitting should be 10.93 μHz, which is 1.7 μHz smaller than the observed splitting. Because of the persistence of this discrepancy, we choose not to consider the 350 s mode splitting to determine the rotation period or as a criterion for selecting the model that fits the best. The 143 s mode typically has a model Cℓ, k-value of about 0.05–0.06, and this yields rotation periods of over 14 hr. In this case, we suspect that we do not correctly predict the mode kinetic energy, and that the 143 s mode should really be a trapped mode with a Cℓ, k ~ 0.13. Because all of the models produce a rotation period from the 143 s mode that is too long, we choose not to include this mode for our rotational splitting constraints. We will comment on the agreement of the rotation periods predicted by the 118, 192, and 114 s modes at each stellar mass, and then discuss the overall quality of the rotation period matches and the best value of the rotation period in our summary.
Finally, we compare the seismological parallax to the observed parallax, which offers an independent check on the luminosity. To compute the seismological parallax, we first convert the blackbody luminosity of our models to MBol by way of
where M⊙Bol = 4.75 (Allen 1973). Then we use the bolometric corrections in Table 1 of Bergeron, Wesemael, & Beauchamp (1995a) to convert MBol to MV = MBol-BC. Typical values are -0.611 mag at 12,000 K and -0.441 mag at 11,000 K. Next, we use the distance modulus formula
to derive the distance in parsecs, using mV = 13.75 for L19-2 and mV = 14.32 for GD 165 (Bergeron et al. 1995b). The Yale parallax catalogue (van Altena et al. 1995) lists the parallax for L19-2 as 44 ± 5 mas. For GD 165, the parallax from the Yale parallax catalogue is 27.8 ± 3.4 mas, and the independent parallax value of 25.4 ± 7.4 mas from Tinney et al. (1995) is consistent with the earlier, more accurate measurement. We will use the Yale parallax catalogue value in our comparisons.
6.1. 0.70 M⊙ Models
Our 0.70 M⊙ models have log g-values of 8.16, which places them at the low end of the 1 σ log g range from spectroscopy. Table 4 shows the periods of selected 0.70 M⊙ models as a function of hydrogen layer mass. The best overall period matches occur for hydrogen layer masses between 1.5 × 10-4 M* and 1.0 × 10-4 M*, although none of the models matches the ℓ = 2 mode at 143 s, and the models predict that the ℓ = 1, k = 6 mode lies between 344 and 354 s.
Table 4 Periods for Various 0.70 M⊙ Models vs. L19-2
We then focus on models with hydrogen layer masses between 1.5 × 10-4 M* and 1.0 × 10-4 M* and independently vary the core composition, along with the composition gradients between the C/O core and the carbon layer, the He/C transition zone, and the H/He transition zone. The nominal C/O core profile fits the observed data best, provided we make the composition gradient at the bottom of the H/He transition zone steeper than the nominal "diffusive equilibrium" composition profile (see Arcoragi & Fontaine 1980) provides (see Fig. 1). One way to physically justify a steeper composition gradient is that there is some nuclear burning going on at the base of the H/He transition zone that is not explicitly modeled in our evolution code. In the white dwarf evolutionary code (WDEC), we mimic this by using exponential profiles to model the diffusion "tails," and we change the thickness of the transition region by changing the value of the exponent. The normal exponent at the base of the H/He transition zone is -1.25, but we also tried -2.25 and -3.25; values less than -3.25 produced no significant change. We label the profiles as "normal" (N, or thick) for -1.25, "moderately thin" (MT) for -2.25, and "thin" (T) for -3.25 (see Fig. 1). The steeper composition gradient affects the periods in the following manner. The ℓ = 1, k = 1 mode becomes about 3–5 s shorter from normal to thin; the ℓ = 1, k = 6 mode becomes about 8–16 s longer from normal to thin; in some cases, the ℓ = 2, k = 4 mode can become about 10 s shorter with a thin transition region, whereas in other cases the period is hardly affected. We find small period differences (less than 2 s) for the k = 2 modes of ℓ = 1 and 2.
Examination of a number of models shows there are three candidate models with overall period errors of less than 3 s (see Table 4), and the best-matching model has an average error of 2.7 s for all five modes. Two of these models have no error larger than 6 s, but there is more spread among the period errors of the modes. The last one is a splendid fit for all but the 143 s mode, but for that mode the error is over 11 s. In all cases, we find the "nominal" C/O core profile provides the best fit. All four of the models have a hydrogen layer mass of 1.5 × 10-4 M* or 1.0 × 10-4 M* and a helium layer mass that is 100 times thicker than the hydrogen layer. All of the models have 20 : 80 C/O cores, and the core extends to 0.75 M*. At the C/O core boundary, there is a linear ramp to a pure carbon composition by 0.90 M*. Four of these models have Teff between 11,450 and 12,100 K, while the fifth is somewhat hotter, at 12,760 K.
An examination of the rotation period implied by the splitting coefficients, Cℓ, k, shows that the model with the best agreement with the rotation period is the one with the very long ℓ = 2, k = 4 period. In this case, the rotation period is 12.96 ± 0.2 hr. We define the rotation period spread as the maximum deviation from the average rotation period derived from the three modes we use. The other models have rotation period spreads of more than ±0.5 hr, although the average rotation period is still about 13 hr.
The luminosity of the best-fitting models is log L/L⊙ = -2.5 to -2.6, which implies a parallax angle of 41–44 mas, well within the error bar of the observed 44 ± 5 mas.
6.2. 0.75 M⊙ Models
The observed spectroscopic log g-value of 8.21 is slightly lower than the log g = 8.24 of our 0.75 M⊙ models. A hydrogen layer mass between 10-4 M* and 8 × 10-5 M* does the best job of matching the observed periods of the 118, 192, and 113 s modes. While good, the period match is not perfect; the period spacing between the ℓ = 1, k = 1 and 2 modes predicted by the models tends to be too short.
One way that we are able to improve the period match is to allow the helium layer mass to be less than 100 times the hydrogen layer mass. In this case, we let the helium layer mass be 6 × 10-3 M* (60 MH), and the hydrogen layer mass is 10-4 M*; the resultant model periods fit to within an average of 2.5 s, and no predicted period deviates by more than 5.3 s from an observed period (see Table 5). The drawbacks of this model are that it has an effective temperature of 11,600 K (too cool) and that our predicted helium layer composition profile never has a pure helium composition. There is always hydrogen, carbon, or both present. Arcoragi & Fontaine (1980) suggest that such a chemical composition profile is not realistic and that diffusion would bring carbon up to the surface, which we do not observe. However, their calculation did not explicitly model the diffusion of the elements in the envelope; to our knowledge, this has not been done for a DA model envelope. We believe it would be worthwhile to redetermine the minimum helium layer mass required to prevent the diffusion of carbon up to the surface.
Table 5 Periods for Various 0.75 M⊙ Models vs. L19-2
Given the problems associated with having a helium layer mass only 60 times the hydrogen layer mass, we study other ways to adjust the chemical composition profile to obtain a good period agreement. We start by fixing the hydrogen layer mass at 8 × 10-5 M* and vary the helium layer mass, C/O core structure, and the steepness of the transition profile in the helium-rich part of the H/He transition zone. Example model fits are shown in Table 5, where we find that the best fit occurs when we use a helium layer mass of 8 × 10-3 M*. We can improve the fit even more by adjusting the exponents that govern the width of the helium-rich part of the H/He transition region to make it steeper; at this point, our best-fitting model matches all five periods with an average error of 1.6 s. The best-fitting model (see Table 5) has a 20 : 80 C/O core out to 0.65 M*, then has a linear ramp to pure carbon by 0.90 M*, a helium layer mass of 8 × 10-3 M*, and a hydrogen layer mass of 8 × 10-5 M*. The temperatures of the best-fitting models are 11,500–11,900 K, however, so they are somewhat cooler than the spectroscopic value of 12,100 ± 250 K. While the period agreement is no longer optimum, models with effective temperatures as high as 12,020 K have overall period matches to within 1.9 s.
We then look at the rotation periods predicted by the Cℓ, k-values to see if they offer additional constraints on which models fit the best. In particular, we are interested in seeing if models with helium layers only 60 times thicker than the hydrogen layer do well in matching the observed frequency splitting. These models produce rotation periods for the three principal modes that match to within 0.5 hr, and the average rotation period is 12.80 hr. Other models can match the rotation period to within 0.6 hr, and the best fits occur with the nominal C/O profile rather than with transition points farther in, even though the model with the C/O transition point located at 0.65 M* fit the observed periods better. This is consistent with what we see for our 0.72 and 0.74 M⊙ models, as discussed in the next subsection.
The luminosity of the best-fitting models is log L/L⊙ = -2.65 to -2.75, which leads to a parallax angle of 46.4–48.3 mas. This seismological parallax is less than 1 σ larger than the observed 44 ± 5 mas. Most of this difference is due to the effective temperature of the best-fitting models dropping to between 11,500 and 11,800 K, which is cooler than the spectroscopic temperature.
6.3. Models With Other Masses
We also calculated grids of models at 0.65, 0.72, 0.74, and 0.78 M⊙. We provide an abbreviated summary of our results for these models, as the general description of the model fits is the same as for the 0.70 and 0.75 M⊙ models.
The calculated log g of our models below 0.70 M⊙ falls below the lower bounds of likely surface gravities, but we examine 0.65 M⊙ models for completeness. Even for an implausibly thick hydrogen layer mass2 of 3.0 × 10-4 M*, the period of the ℓ = 1, k = 1 mode is more than 121 s, or about 3 s higher than the observed period for a model at 11,940 K. To double-check the validity of these models, we would need to include the ability to model diffusion-induced pp burning at the base of the hydrogen layer, which WDEC does not do. This model has the interesting feature that the Cℓ, k-values for all five modes produce rotation periods that agree with each other to within 0.6 hr, and the average rotation period is 12.98 hr. The agreement is most likely coincidental, because the periods do not match the observations as well as other models can.
The 0.65 and 0.70 M⊙ models show a trend of decreasing hydrogen layer mass with increasing stellar mass (see Fig. 3), but from 0.72 to 0.78 M⊙, we see very little change in the optimum hydrogen layer mass, which lies between 8 × 10-5 M* and 1.2 × 10-4 M*. The favored helium layer mass trend follows the same trend as the hydrogen layer mass. The only difference is that the helium layer mass is typically 100 times thicker than the hydrogen layer mass.
Fig. 3 Best-fitting hydrogen layer mass vs. stellar mass for models that fit L19-2 well, with effective temperature. Note the small-range (factor of 4) variation in hydrogen layer mass with stellar mass. The 0.65–0.74 M⊙ models provide reasonable fits to the effective temperature. When we consider period matching, parallax, effective temperature, and rotational splitting, the 0.72 M⊙ model fits best.
The 20 : 80 C/O core mass fraction is 0.60 M* at 0.72 M⊙ and 0.65 M* at 0.74 and 0.78 M⊙. The optimum hydrogen layer mass lies at 10-4 M* for the 0.72 and 0.78 M⊙ models, while the 0.74 M⊙ models have a best-fitting hydrogen layer mass of 8 × 10-5 M*. Likewise, there is a narrow range of helium layer masses, ranging from 10-2 M* (0.72 and 0.78 M⊙) to 8 × 10-3 M* at 0.74 M⊙. Finally, the implied rotation periods are the same, with values between 13.0 and 13.1 hr.
The big difference for the different mass models is that the effective temperature of the best-fitting model and the resultant luminosity (and parallax angle) decrease with increasing mass. At 0.74 M⊙ and below, the effective temperature of the best models lies within the ±250 K error bars, while the 0.78 M⊙ model with the best matches to the observed periods has an unrealistically low value of 10,100 K. As a result, the luminosity of the 0.72 and 0.74 M⊙ models is log L/L⊙ = -2.6 to -2.7, which implies a parallax angle of 43.8–46.5 mas, well within 1 σ of the observed 44 ± 5 mas. At 0.78 M⊙, the luminosity is log L/L⊙ = -2.86 to -3.0, which leads to a parallax angle of 57–60 mas, too large by 2–3 σ.
2 Calculations from a number of sources (Iben & MacDonald 1986; Koester & Schönberner 1986; Hansen 1999) suggest that nuclear burning and other processes should limit the hydrogen layer mass to about 10-4 M* at 0.60 M⊙, and that the hydrogen layer mass becomes thinner with increasing stellar mass.
6.4. Summary of Model Fits for L19-2
Here, we describe the properties of our best-fitting model, which has a mass of 0.72 M⊙ (see Table 6 for a summary of the period fits for the best-fitting models of each mass). Based on our models, the realistic mass range for L19-2 is from 0.70 to 0.75 M⊙. When we consider the quality of the period fit, matching the effective temperature, and matching the observed parallax, the 0.72 M⊙ models clearly do the best job. The effective temperature matches to within the ±250 K error bars, the log g-value of 8.19 is only 0.02 dex lower than the observed value, and the seismological parallax of 44 mas is an exact match to the observed value of 44 ± 5 mas (Bergeron et al. 1995b). The hydrogen layer mass is 10-4 M* and the helium layer mass is between 9 × 10-3 M* and 10-2 M*. The 20 : 80 C/O core extends out to 0.60 M*, with a linear ramp from there to pure carbon by 0.90 M*.
Table 6 Summary of Best-fitting Models for L19-2
All the best-fitting models predict very similar rotation periods for L19-2, clustering around 12.9–13.0 hr with a spread of 0.2–0.3 hr in the best cases. However, the frequency splittings for the 350 and 143 s modes do not fit this pattern (with the exception of the 0.65 M⊙ model that has an overly thick hydrogen layer). The 350 s mode has a splitting so close to the 1 day alias peak that we are not sure if the observed splitting is correct (but see O'Donoghue & Warner 1982). In the case of the 143 s mode, we suspect that the kinetic energies of our models are incorrect. Our theoretical C2, 4-values are close to 0.05, and the mode is strongly nontrapped. The C2, 4-value required for a ~13 hr rotation period is closer to 0.135, which would be appropriate for a trapped mode. Thus, the rotational splitting information suggests that our models need more improvement to be a perfect match to the data. We suspect that most of the improvements need to be made to the input physics, because the period (the eigenvalue) matches well, but the eigenfunction appears not to. Another possibility is the formulation of the pulsation equations themselves. We iterate to convergence for the eigenvalue and do not have a convergence criterion for the eigenfunction, which means that there could be errors in the eigenfunction affecting the kinetic energy even when we have a converged eigenvalue. We still need better observations to confirm some of the critical modes and frequency splittings; then we need to look at improving the models so we can learn more about the physics of pulsating white dwarf stars.
6.5. Predicted Rates of Period Change for L19-2
In principle, the rate of period change for a given mode can provide an estimate of the evolutionary cooling timescale of a white dwarf. O`Donoghue & Warner (1987) provide an upper limit for the rate of period change of the 192 s mode of L19-2 that is less than 3 × 10-14 s s-1, and Sullivan et al. (2001) should be providing a better estimate soon based on over 20 years of data. As part of our analysis of the theoretical models, we provide calculated rates of period change for comparison to the 192 s mode and show the expected values for all the other observed modes in Table 7. We caution that our theoretical rates of period change do not take into account any effects due to rotation or relatively rapid timescale processes, such as time-varying magnetic fields.
Table 7 Rates of Period Change for Selected Models that Match L19-2
Table 7 shows calculated rates of period change for the five observed modes from the best-fitting models of each stellar mass we examine. One general trend is immediately apparent, and that is a general decline in the calculated rate of period change with increasing stellar mass; this just reflects the greater overall heat capacity of a more massive white dwarf. For our most favored mass of 0.72 M⊙, we predict that the 192 s mode should have a rate of period change of 2.0 to 2.2 × 10-15 s s-1, which is an order of magnitude smaller than the upper limit of O'Donoghue & Warner (1987). This disagreement is no surprise, as the data only spanned 8 years in that data set. Sullivan et al.'s (2001) data set is 3 times as long, so its errors should be about 10 times smaller.
Of the other modes, clearly the 350 s one has the largest predicted value. However, the 350 s mode has a low amplitude, so obtaining good phase information for it will be difficult. The other three modes have similar rates of period change (~1.5 × 10-15 s s-1), and they are all smaller than what we estimate for the 192 s mode. The reason for the change in is that /P provides the cooling-timescale estimate, which is similar for all periods. The nearly constant /P-value implies that shorter period modes should have smaller -values, which is what we see.
7. STRUCTURAL CONSTRAINTS FROM ALL PERIODS FOR GD 165
Fitting the 120 s period constrains the hydrogen layer mass to be near 10-4 M*, and the log g-value suggests that the stellar mass is at least 0.65 M⊙. We have two rotationally split modes and two modes with single peaks for GD 165, listed in Table 2, that we can compare with our model predictions. As we did for L19-2, we perform our analysis in two parts; first, we discuss the period fitting for models of different structures, then we discuss the quality of the period matches, the chemical composition profile that is required, and the uncertainties in the profile. We examine models with masses of 0.65, 0.68, 0.70, 0.72, and 0.74 M⊙. Although this does not exhaust all of the possible models that might fit GD 165, the grid is dense enough to determine the trend of the chemical composition profile as a function of stellar mass. The number of models we compute for a given stellar mass varies. From our work on L19-2, our original model grid had models of 0.65, 0.70, 0.75, and 0.80 M⊙. We computed large numbers of models for these masses; later, we looked at the intermediate masses after we already had a good idea of the chemical composition profile that would produce the best period match. Therefore, we computed relatively few of these models.
In addition to the period matches, we use the m-splitting information for the 120 and 192 s modes as additional constraints on the model fits. The splittings for each triplet from the complete data set of Bergeron et al. (1993) are nonuniform. For simplicity, we simply average for each mode the frequency splits between the m = 0 and +1 mode and the m = 0 and -1 mode. For the 120 s mode, the two splittings are 2.73 and 2.55 μHz, and the average value is 2.64 μHz. For the 192 s mode, the two splittings are 2.98 and 2.82 μHz, and the average value is 2.90 μHz. However, this is probably not the most accurate procedure; one should consult Bergeron et al. (1993) for a detailed discussion of the fine-structure splitting and its dependence on the size of the data set used. Finally, we also attempt to match the observed parallax, which offers an independent check on the luminosity. We use mV = 14.32 for GD 165 (Bergeron et al. 1995b). For GD 165, the parallax from the Yale parallax catalogue (van Altena et al. 1995) is 27.8 ± 3.4 mas, and we use this in our comparisons.
7.1. 0.65 M⊙ Models
GD 165 has a spectroscopic log g of 8.06, which is nearly identical to the log g-values of 8.07 from our 0.65 M⊙ models. Taking a cue from our work on L19-2, we use a thin H/He transition zone base to bring down the period of the ℓ = 1, k = 1 mode. With this, we are able to obtain good period matches with a hydrogen layer mass of 2.0 × 10-4 M* and a helium layer mass of 1.9 or 2.0 × 10-2 M* (see Table 8). Models with hydrogen layer masses of 1.8 or 2.2 × 10-4 M* and models with helium layer masses of 1.8 or 2.2 × 10-2 M* do not fit the observed periods as well (see Table 8). Our best-fitting model has a C/O core that extends out to 0.55 M*, with a linear transition to pure carbon by 0.90 M*. Here too, models with slightly different parameters (core masses of 0.50 or 0.60 M*) do not fit the observations as well. The best-fitting model matches all four periods with an average error of 0.75 s. The effective temperature of the best-fitting model is about 12,200 K, which is just within the upper part of the error bar for the observed 11,980 ± 250 K. The seismological parallax turns out to be 31.2 mas, which is 1 σ greater than the observed parallax angle. The 0.65 M⊙ models also happen to be the hottest and most luminous models we examine, so the parallax angle discrepancy will be worse with the higher-mass models. It is also possible that the observed parallax angle is underestimated by more than 1 σ. We feel we can confidently suggest that the mass cannot be significantly lower than 0.65 M⊙, simply because the hydrogen layer mass required to obtain an ℓ = 1, k = 1 period of 120 s becomes impossibly thick because of pp-burning constraints at the bottom of the hydrogen layer. This restriction rules out the Teff = 11,800 ± 330 K and log g = 7.76 ± 0.35, solution for the atmospheric parameters of GD 165 obtained by Koester & Holberg (2001) using the observed parallax angle as the added constraint for fitting the UV spectrum.
Table 8 Periods for Various 0.65 M⊙ Models vs. GD 165
An examination of the Cℓ, k-values for the best-fitting models shows that C1, 1 = 0.474 and C1, 2 ranges from 0.355 to 0.369. The implied rotation period for k = 1 mode is 55.3 hr, while the k = 2 mode has a rotation period between 60.4 and 61.8 hr. This disparity shows up for all the models we examine. Bergeron et al. (1993) have a detailed discussion of the fine-structure splitting and of how the values differ depending on whether one uses all of the available data or just the Whole Earth Telescope run data. Here, we discuss several theoretical ways to resolve this discrepancy. First, we use the average Cℓ, k-values from our models and each individual frequency splitting to determine the range of implied rotation periods. Using C1, 1 = 0.474 and assuming that the splitting is 2.55 or 2.73 μHz, we find the rotation period can be either 57.3 or 53.2 hr. We use an average value of C1, 2 = 0.361 and derive rotation periods of 59.6 and 62.9 hr, for splittings of 2.98 and 2.82 μHz, respectively. So, even if we use the most favorable observed frequency splittings, we still have a difference of over 2 hr in the implied rotation period. The second approach that we take is to assume that the C1, 1 = 0.474 value is absolutely correct (and C1, 1 shows little variation for all the models for GD 165 that we examine), take the implied rotation period of 55.3 hr, and determine the value of C1, 2 required to produce the same 55.3 hr rotation period. This exercise produces C1, 2 = 0.423, which would suggest that the k = 2 mode should have a lower kinetic energy than the models calculate. One cannot appeal to second-order effects to explain the difference between the m = +1 to m = 0 and the m = -1 to m = 0 splittings, because the observed nonuniformities are about 0.2 μHz, far larger than the 0.001 μHz nonuniformities seen in the frequency splittings of L19-2 that Brassard, Wesemael, & Fontaine (1989) attribute to second-order splitting based on model calculations. About all we can conclude concerning the discrepancy between the implied rotation periods for the k = 1 and 2 modes is that we need to resolve the frequency spacing problem observationally before we can be sure there is a "problem" with the predicted Cℓ, k-values from models. While we do not go through such a detailed exercise for the other masses, we note that the above discussion applies equally well for the other models.
7.2. Models With Masses Between 0.68 and 0.74 M⊙
Our discussion of the period matching and problems with matching the observed frequency splittings at 0.65 M⊙ is relevant to other models with masses between 0.68 and 0.74 M⊙. The main trend is that the log g-value and implied parallax angle grow larger with increasing model mass, and the fit to observations becomes worse as a result. However, as we mention above, taking the observed parallax angle at face value pushes one to low enough stellar masses that the required hydrogen layer mass becomes unrealistically thick. Therefore, we feel it is inappropriate to place too much weight on the parallax angle in discriminating between models.
For each mass, the best-fitting models match all four observed periods to an average error between 0.5 and 1.0 s, with the main differences being the size of the C/O core and effective temperature. There is a jump in the size of the C/O core between 0.68 and 0.70 M⊙. Below 0.70 M⊙, the preferred core mass is 0.55–0.65 M*, while a 0.75 M* core provides the best period match at 0.70 M⊙ and above. This trend runs counter to that seen for L19-2, where the core mass fraction decreases with increasing stellar mass. Interestingly, the best-fitting models for each star have a similar C/O core mass fraction of about 0.60 M*. The jump in the C/O core size also causes a jump in the effective temperature of the best-fitting model. At 0.68 M⊙ the effective temperature is 11,900 K, which jumps to almost 12,600 K at 0.70 M⊙. The effective temperature of the best-fitting models drops after that, down to 11,800 K at 0.72 M⊙ and only 11,320 K at 0.74 M⊙.
The hydrogen layer mass that produces the best period match spans a narrow range, from 1.5 × 10-4 M* at 0.68 M⊙ to 8 × 10-5 M* at 0.70 and 0.74 M⊙. The helium layer is typically 100–125 times more massive than the hydrogen layer. Both the hydrogen and helium layer masses are well constrained; changes in either layer mass of 10%–20% produce noticeably worse fits to the observed periods. The best-matching models to GD 165 require a thin (or sometimes moderately thin) H/He transition zone base. Again, this is similar to L19-2, where we also require a thinner-than-normal H/He transition zone base for the best period match.
The Cℓ, k features noted in the 0.65 M⊙ models show up for all the other masses as well. The value of C1, 1 lies between 0.477 and 0.481, while C1, 2 lies between 0.341 and 0.381. In both cases, there is a slight trend toward larger values of Cℓ, k at larger masses. These rotational splitting coefficients translate into rotation periods between 54.5 and 55.0 hr for the k = 1 mode and between 59.3 and 63.1 hr for the k = 2 mode. The 0.70 M⊙ model has the smallest discrepancy, 4.4 hr, with rotation periods of 54.9 hr and 59.3 hr for the k = 1 and 2 modes. The relatively smaller rotation period disagreement appears to be due to changes in mode trapping brought on by increasing the size of the C/O core. If we increase the C/O core mass fraction to 0.80 M*, the rotation period difference drops further to 3.7 hr, and the relative kinetic energies of the k = 1 and 2 modes flip, so that the k = 2 mode has the lower kinetic energy. This improved agreement in the rotation period comes at the expense of the period agreement.
7.3. Summary of Model Fits for GD 165
Our results show that GD 165 is indeed quite similar in structure to L19-2, with the biggest differences being the mass fraction of the C/O core and the best-fitting stellar mass (see Table 9 for a summary of best-fitting models as a function of stellar mass). Based on our models, we believe that the realistic mass range for GD 165 is about 0.65–0.70 M⊙, although none of our models matches the observed parallax values to better than 1 σ. However, we feel that the parallax angle discrepancy may be more the result of the true parallax angle being at least 1 σ greater than the observed values, mainly because the required hydrogen layer mass becomes unrealistically thick if we make the total mass of the model low enough to precisely match the observed parallax.
Table 9 Summary of Best-fitting Models for GD 165
At this point, we feel the best-fitting model has a mass of 0.65 M⊙; it matches all four periods to an average of 0.75 s. The effective temperature matches to within the ±250 K error bars, the log g-value of 8.06 is only 0.01 dex lower than the observed value, and the seismological parallax of 31.2 mas is about 1 σ greater than the observed value of 27.8 ± 3.4 mas (van Altena et al. 1995). However, Koester & Holberg (2001) derive reasonable values of Teff = 12,310 ± 180 K and log g = 8.20 ± 0.15 when they use the V magnitude as the added constraint for fitting the HST UV spectra with model atmospheres. These values of Teff and log g would imply that the observed parallax angle should be about 34 mas.
The hydrogen layer mass is 2 × 10-4 M*, and the helium layer mass is between 1.9 × 10-2 M* and 2.0 × 10-2 M*. The 20 : 80 C/O core extends out to between 0.55 and 0.60 M*, with a linear ramp from there to pure carbon by 0.90 M*. We emphasize that these values are relatively insensitive to the stellar mass of the best-fitting model. The hydrogen layer mass lies between 0.8 and 2.0 × 10-4 M* (see Fig. 4), while the helium layer is about 100 times more massive than the hydrogen layer for all the best-fitting models between 0.65 and 0.74 M⊙. The C/O core mass fraction is a bit more sensitive, with the 20 : 80 C/O core extending to between 0.55 and 0.75 M* depending on whether the mass is 0.65 M⊙ or 0.75 M⊙.
Fig. 4 Same as Fig. 3, but for GD 165. Note the factor of 3 variation in hydrogen layer mass with stellar mass. The 0.65, 0.68, and 0.72 M⊙ models provide reasonable fits to the effective temperature. When we also consider the period and parallax, we favor the 0.65–0.68 M⊙ models.
The observed frequency splittings for GD 165 are problematic to interpret. We typically predict a rotation period of about 55 hr for the 120 s mode, while the rotation period of the 192 s mode tends to be longer (59–63 hr) and has a larger spread. The best rotation period agreement comes from a 0.70 M⊙ model, where the periods are 54.7 hr for the 120 s mode and 58.4 hr for the 192 s mode, for a difference of 3.7 hr. If we take a rotation period of 54–55 hr (predicted by the 120 s mode) as accurate, we would require that the C1, 2-value be closer to 0.42 or 0.43 than to the calculated 0.36–0.39. Taken at face value, this would imply that our models are predicting too large a kinetic energy value for the 192 s mode. However, the observational picture is confusing enough that it is not obvious what conclusion can be drawn other than that the rotation period is somewhere in the 55–60 hr range and that more observational and theoretical work is needed.
The uncertainty in what the best-fitting model might be for GD 165 underscores two needs. First, we could really use a better determination of the parallax angle for GD 165; the model-atmosphere parameters and the restrictions on the maximum allowable hydrogen layer mass point to a larger parallax angle than the one quoted by van Altena et al. (1995). Second, assuming that we can obtain a more accurate parallax angle, we then need to rederive the best model-atmosphere parameters that fit the UV spectrum of GD 165. Then we can work through our seismological period-fitting exercise to determine the best-fitting model.
7.4. Predicted Rates of Period Change for GD 165
There are no observed estimates of the rate of period change for any of the modes of GD 165, but Table 10 provides our theoretical predictions for the observed modes. The 192 s mode of GD 165 is expected to have a rate of period change of 2.6 to 2.8 × 10-15 s s-1, which is about 20% larger than what we predict for L19-2, mainly because of the somewhat lower stellar mass we favor for GD 165. Table 10 shows that the largest rate of period change occurs for the longest period (smallest amplitude) mode. Given the low amplitude of the 250 s mode, we favor trying to detect a rate of period change for the 192 s mode. Detecting the rate of period change will be a challenging observational project, as previous experience with L19-2 and G117-B15A shows that more than 20 years' worth of accurate data are needed. Add to this the fact that the 192 s mode is a triplet with a frequency spacing of about 2.8 μHz, which means that we need about a week's worth of data for each data set to be useful. Still, given the dearth of stars with rate of period change estimates, observers should at least start such a program to see if there are surprises.
Table 10 Rates of Period Change for Selected Models that Match GD 165
8. SUMMARY AND CONCLUSIONS
Combined with the hydrogen layer mass determinations for other DA white dwarfs (see Table 1), it appears that hydrogen layer masses near 10-4 M* are preferred for ZZ Ceti stars in the small sample studied thus far. If this hydrogen layer mass trend is typical for DA white dwarfs, then the standard picture of stellar evolution from the asymptotic giant branch (AGB) to the white dwarf stage (Iben & MacDonald 1986; D'Antona & Mazzitelli 1991) is at least qualitatively correct. However, GD 154 and G29-38 could be objects that have a slightly different mass-loss history on the AGB; the hydrogen layer mass is sensitive to the phase during the thermal flash cycle, when the final envelope ejection takes place.
The underlying helium layer is close to 10-2 M* for both stars, mainly because the hydrogen layer is about 10-4 M* and the helium layer must be at least ~100 times thicker (Arcoragi & Fontaine 1980). Less massive helium layers should allow carbon to diffuse to the surface, in violation of spectroscopic observations. However, to our knowledge, the lower limit to how massive the helium layer can be has not yet been explored with evolution/diffusion calculations, and this effort would be worthwhile, as we found some models that fit the L19-2 periods reasonably well with a helium layer mass only 60 times greater than the hydrogen layer mass. Helium layer masses much greater than 10-2 M* are not allowed because of the problem of (explosive) helium burning that would be present.
Both GD 165 and L19-2 appear to have oxygen-rich cores, in common with at least G117-B15A and R548 (constraints on the other stars in Table 1 are weaker). The C/O ratio in the core is the same (20 : 80) for both L19-2 and GD 165. For the most favored masses, the extent of the core is about 0.55–0.65 M*. Our results suggest that the 12C(α, γ)16O rate is at least the value of Caughlan & Fowler (1988) and is consistent with the reaction rate of Caughlan et al. (1985).
Both stars have at least some modes that exhibit fine-structure splitting consistent with rotation, and we derive rotation rates for each based on the first-order (Cℓ, k) splitting coefficients. Both stars present complications in determining the rotation rate, in that the value is strongly mode-dependent. In the case of L19-2, the three dominant (118, 192, and 113 s) modes can yield nearly the same result, which lies between 12.8 and 13.0 hr. The 350 and 143 s modes typically yield rotation periods of 12 and 14 hr, respectively. The splitting of the 350 s mode is very close to that expected for a 1 cycle per day alias, although O'Donoghue & Warner (1982) show evidence that the 348 s mode is not an alias. However, it may be affected at the level of a few tenths of a microhertz, which would be enough to provide agreement. We suspect that our models produce the wrong kinetic energy for the 143 s mode, in turn producing a calculated C2, 4-value that is too low. In addition to having different rotation periods predicted by the fine-structure splitting of the 120 and 192 s modes, we also have the problem of how to interpret the observed nonuniform fine-structure splitting. Until we have better observations feeding data into improved models, we can only say that the rotation period is about 55–60 hr.
Finally, there is the matter of the parallax. Both L19-2 and GD 165 have seismological parallax values that agree to within 1 σ or better with the best available trigonometric parallax values, in contrast to the profound disagreement for G117-B15A and R548 (Bradley 1998b). Part of the difference in agreement is likely due to the fact that L19-2 and GD 165 have much larger parallax values, at 44 and 28 mas, compared to values below 15 mas for G117-B15A and R548. The gross seismological and trigonometric parallax disagreements at small parallax angles reinforce our need for accurate parallax determinations for more distant white dwarfs, if it is possible to obtain them. Future satellite parallax missions like Gaia and ground-based CCD parallax surveys with at least 2 m telescopes offer the best hope for producing the requisite data. Our work also reinforces the value of seismological parallaxes, which offer an independent method of estimating the distance to white dwarf stars.
Our results for L19-2 and GD 165 brings to five (out of seven) the number of white dwarfs with favored hydrogen layer masses in the neighborhood of 10-4 M*. These results suggest that ZZ Ceti white dwarfs have hydrogen layer masses that can be consistent with most DA white dwarfs having either "thick" hydrogen layers or a range of hydrogen layer masses. Indirect evidence from other sources argues for a range of hydrogen layer masses in DA white dwarfs, starting with Vennes et al. (1997) for the hotter DA's, where hydrogen layer masses of 10-6 M* or thinner are required for their mean white dwarf mass to match the mean mass of the cooler Bergeron, Saffer, & Liebert (1992) sample. Also, Bergeron et al. (1990) show that there is extra pressure in DA atmospheres starting at about 10,000 K, which can be interpreted as helium being "dredged up" to the surface of DA white dwarfs with hydrogen layers near 10-10 M*. Finally, there is the work of Bergeron, Ruiz, & Leggett (1997) for the coolest white dwarfs, where the observations are also best interpreted when there is a range of hydrogen layer masses. We still need more seismological hydrogen layer mass values to understand better what the hydrogen layer mass distribution might be and also to provide confirmation (or glaring inconsistencies) that will help guide future theoretical modeling efforts.
Hansen (1999) presents hydrogen layer masses as a function of white dwarf mass, which we compare to the hydrogen layer masses for L19-2, GD 165, and the other ZZ Ceti stars in Table 1. Hansen's (1999) hydrogen layer masses start off at 9.9 × 10-5 M* at 0.6 M⊙, then drop to 9.4 × 10-5 M* at 0.7 M⊙ and to 8.0 × 10-5 M* at 0.8 M⊙. Our results suggest that most of the hydrogen layer masses range from 2.0 × 10-4 M* to 1.0 × 10-4 M* between 0.54 and 0.80 M⊙, without a clear trend of hydrogen layer mass with different stellar masses. This comparison suggests that our hydrogen layer masses are up to a factor of 2 more massive than Hansen (1999) calculates. However, the range in hydrogen layer masses from other calculations shows a factor of 4 spread in hydrogen layer mass at 0.6 M⊙, with a high value of 1.6 × 10-4 M* from Koester & Schönberner (1986) to a low value of 4.1 × 10-5 M* from D'Antona & Mazzitelli (1979). A comparison of our seismologically derived hydrogen layer masses with the above results suggests that while our hydrogen layer masses are higher than the latest values, they are not unreasonable. Finally, Koester & Schönberner (1986) provide the only hydrogen layer mass limit for a 0.55 M⊙ white dwarf